Connecting thorugh a VPN does not provide anonymity. It just provides a different IP address. Any way in which you could otherwise be traced you still be traced.
If you want your VPN-redirected traffic to not be at all associated with your non-VPN-redirected traffic, then your best solution is to keep completely separate profiles for each use. The most complete way to do so is with two separate computers; the second most complete way is by using a separate OS on the same computer (also not that bad), and the third way is to separate out the browsers (not ideal, but less friction).
And then, of course, never do anything within the one profile that you do with the other. No shared accounts, no shared logins, no shared usage patterns, no shared anything. Don't even visit the sames sites.
Often when people's anonymous personae are tracked down (e.g. by a repressive government, etc.) and tied to a real-world person, it's by discovering patters in behavior, speech, timing, or other such side-channel details, or one one persona makes some reference to the other, or by both sharing some common offline characteristic (like the sound of your voice if someone talks to you). The more a given identity does, the more difficult it becomes for that identity to remain secret. The only safety (if there is any at all) is in keeping the anonymous persona sparse and very rarely encountered. It's an extraordinarily dangerous line to tread, so do so with the utmost caution.
Also, and here's a technical detail you don't want to forget: If a "content of interest" is seen by the government coming form a specific IP address, and YOU are seen connecting to a VPN at that same address... well then your secret is not so secret. If your only protection is a VPN, then you're not particularly protected.
